Translation:The Greek actresses are playing in the other city.
Is "playing" used here to mean "performing (on stage)", or are they having a game of something in the park?
In English, "playing" can have both meanings (eg "'Hamlet' is playing at the Royal Theatre until May 5, tickets still available" or "Geoffrey Rush is playing King Lear in the new production") but I wouldn't like to assume Hungarian does the same.
Yes, Hungarian does the same. It could be a game of something, we just assume it is about a performance.
Actually I think that the whole expression "playing in the other city" is incorrect in GB English. Actors and actresses "perform" rather than "play" and stage productions "play" in other locations. This brings into question the use of the verb "játsz". A better alternative would appear to be ""előad" (Just a suggestion as I am not Hungarian. Another word may be better)!. Also "the other" is also incorrect. As the city is not identified, the correct expression is "in another city". "The" is completely unnecessary. The correct translation for this should be "The Greek actresses are performing in another city"
It's a sentence that would make more sense, but it's not a translation of the Hungarian sentence.
As discussed above, játszik can be translated as "perform", so that part of your argument is okay. A more popular possibility is fellép. Előad, on the other hand, is about what is being performed/shown, not who does it.
"A másik város" is indeed "the other city". It has the definite article a in front of it, so it is a very certain city that's being talked about. I doesn't make a lot of sense out of context, of course. Granted, it doesn't even make a lot of sense in context either. But if you already have established a list of two possible cities within the context, it's a very possible expression. Maybe you forgot the name of the other city? It's certainly not Budapest where they're performing tonight.
"Another city" would simply be "(egy) más város".
As with a lot of these "weird" sentences, it could work this way in certain contexts. For instance, if someone asked "what are the actresses doing in the other city?" then "the actresses are playing in the other city" would be a correct (if wordy) response.